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Theorem cnextf 20296
Description: Extension by continuity. The extension by continuity is a function. (Contributed by Thierry Arnoux, 25-Dec-2017.)
Hypotheses
Ref Expression
cnextf.1  |-  C  = 
U. J
cnextf.2  |-  B  = 
U. K
cnextf.3  |-  ( ph  ->  J  e.  Top )
cnextf.4  |-  ( ph  ->  K  e.  Haus )
cnextf.5  |-  ( ph  ->  F : A --> B )
cnextf.a  |-  ( ph  ->  A  C_  C )
cnextf.6  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  C )
cnextf.7  |-  ( (
ph  /\  x  e.  C )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =/=  (/) )
Assertion
Ref Expression
cnextf  |-  ( ph  ->  ( ( JCnExt K
) `  F ) : C --> B )
Distinct variable groups:    x, A    x, B    x, C    x, F    x, J    x, K    ph, x

Proof of Theorem cnextf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cnextf.3 . . . 4  |-  ( ph  ->  J  e.  Top )
2 cnextf.4 . . . 4  |-  ( ph  ->  K  e.  Haus )
3 cnextf.5 . . . 4  |-  ( ph  ->  F : A --> B )
4 cnextf.a . . . 4  |-  ( ph  ->  A  C_  C )
5 cnextf.1 . . . . 5  |-  C  = 
U. J
6 cnextf.2 . . . . 5  |-  B  = 
U. K
75, 6cnextfun 20294 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Haus )  /\  ( F : A --> B  /\  A  C_  C
) )  ->  Fun  ( ( JCnExt K
) `  F )
)
81, 2, 3, 4, 7syl22anc 1224 . . 3  |-  ( ph  ->  Fun  ( ( JCnExt
K ) `  F
) )
9 simpl 457 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  ph )
10 cnextf.6 . . . . . . . . 9  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  C )
1110eleq2d 2532 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( ( cls `  J
) `  A )  <->  x  e.  C ) )
1211biimpar 485 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  ( ( cls `  J
) `  A )
)
13 cnextf.7 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =/=  (/) )
14 n0 3789 . . . . . . . 8  |-  ( ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =/=  (/)  <->  E. y 
y  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { x } )t  A ) ) `  F ) )
1513, 14sylib 196 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  E. y 
y  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { x } )t  A ) ) `  F ) )
16 haustop 19593 . . . . . . . . . . . . . 14  |-  ( K  e.  Haus  ->  K  e. 
Top )
172, 16syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  Top )
185, 6cnextfval 20292 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  C
) )  ->  (
( JCnExt K ) `
 F )  = 
U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
191, 17, 3, 4, 18syl22anc 1224 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( JCnExt K
) `  F )  =  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
2019eleq2d 2532 . . . . . . . . . . 11  |-  ( ph  ->  ( <. x ,  y
>.  e.  ( ( JCnExt
K ) `  F
)  <->  <. x ,  y
>.  e.  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
21 opeliunxp 5045 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  <->  ( x  e.  ( ( cls `  J
) `  A )  /\  y  e.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
2220, 21syl6bb 261 . . . . . . . . . 10  |-  ( ph  ->  ( <. x ,  y
>.  e.  ( ( JCnExt
K ) `  F
)  <->  ( x  e.  ( ( cls `  J
) `  A )  /\  y  e.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
2322exbidv 1685 . . . . . . . . 9  |-  ( ph  ->  ( E. y <.
x ,  y >.  e.  ( ( JCnExt K
) `  F )  <->  E. y ( x  e.  ( ( cls `  J
) `  A )  /\  y  e.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
24 19.42v 1944 . . . . . . . . 9  |-  ( E. y ( x  e.  ( ( cls `  J
) `  A )  /\  y  e.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  <->  ( x  e.  ( ( cls `  J
) `  A )  /\  E. y  y  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
2523, 24syl6bb 261 . . . . . . . 8  |-  ( ph  ->  ( E. y <.
x ,  y >.  e.  ( ( JCnExt K
) `  F )  <->  ( x  e.  ( ( cls `  J ) `
 A )  /\  E. y  y  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
2625biimpar 485 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( cls `  J
) `  A )  /\  E. y  y  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )  ->  E. y <. x ,  y >.  e.  ( ( JCnExt K ) `
 F ) )
279, 12, 15, 26syl12anc 1221 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  E. y <. x ,  y >.  e.  ( ( JCnExt K
) `  F )
)
2825simprbda 623 . . . . . . 7  |-  ( (
ph  /\  E. y <. x ,  y >.  e.  ( ( JCnExt K
) `  F )
)  ->  x  e.  ( ( cls `  J
) `  A )
)
2911adantr 465 . . . . . . 7  |-  ( (
ph  /\  E. y <. x ,  y >.  e.  ( ( JCnExt K
) `  F )
)  ->  ( x  e.  ( ( cls `  J
) `  A )  <->  x  e.  C ) )
3028, 29mpbid 210 . . . . . 6  |-  ( (
ph  /\  E. y <. x ,  y >.  e.  ( ( JCnExt K
) `  F )
)  ->  x  e.  C )
3127, 30impbida 829 . . . . 5  |-  ( ph  ->  ( x  e.  C  <->  E. y <. x ,  y
>.  e.  ( ( JCnExt
K ) `  F
) ) )
3231abbi2dv 2599 . . . 4  |-  ( ph  ->  C  =  { x  |  E. y <. x ,  y >.  e.  ( ( JCnExt K ) `
 F ) } )
33 dfdm3 5183 . . . 4  |-  dom  (
( JCnExt K ) `
 F )  =  { x  |  E. y <. x ,  y
>.  e.  ( ( JCnExt
K ) `  F
) }
3432, 33syl6reqr 2522 . . 3  |-  ( ph  ->  dom  ( ( JCnExt
K ) `  F
)  =  C )
35 df-fn 5584 . . 3  |-  ( ( ( JCnExt K ) `
 F )  Fn  C  <->  ( Fun  (
( JCnExt K ) `
 F )  /\  dom  ( ( JCnExt K
) `  F )  =  C ) )
368, 34, 35sylanbrc 664 . 2  |-  ( ph  ->  ( ( JCnExt K
) `  F )  Fn  C )
3719rneqd 5223 . . 3  |-  ( ph  ->  ran  ( ( JCnExt
K ) `  F
)  =  ran  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
38 rniun 5409 . . . 4  |-  ran  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  = 
U_ x  e.  ( ( cls `  J
) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )
39 vex 3111 . . . . . . . . 9  |-  x  e. 
_V
4039snnz 4140 . . . . . . . 8  |-  { x }  =/=  (/)
41 rnxp 5430 . . . . . . . 8  |-  ( { x }  =/=  (/)  ->  ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  =  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )
4240, 41ax-mp 5 . . . . . . 7  |-  ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  =  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )
4311biimpa 484 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( cls `  J
) `  A )
)  ->  x  e.  C )
446toptopon 19196 . . . . . . . . . . 11  |-  ( K  e.  Top  <->  K  e.  (TopOn `  B ) )
4517, 44sylib 196 . . . . . . . . . 10  |-  ( ph  ->  K  e.  (TopOn `  B ) )
4645adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  C )  ->  K  e.  (TopOn `  B )
)
475toptopon 19196 . . . . . . . . . . . 12  |-  ( J  e.  Top  <->  J  e.  (TopOn `  C ) )
481, 47sylib 196 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  (TopOn `  C ) )
4948adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  C )  ->  J  e.  (TopOn `  C )
)
504adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  C )  ->  A  C_  C )
51 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  C )
52 trnei 20123 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  C )  /\  A  C_  C  /\  x  e.  C )  ->  (
x  e.  ( ( cls `  J ) `
 A )  <->  ( (
( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) ) )
5352biimpa 484 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  C )  /\  A  C_  C  /\  x  e.  C )  /\  x  e.  ( ( cls `  J
) `  A )
)  ->  ( (
( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) )
5449, 50, 51, 12, 53syl31anc 1226 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  C )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) )
553adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  C )  ->  F : A --> B )
56 flfelbas 20225 . . . . . . . . . . 11  |-  ( ( ( K  e.  (TopOn `  B )  /\  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  /\  F : A --> B )  /\  y  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  -> 
y  e.  B )
5756ex 434 . . . . . . . . . 10  |-  ( ( K  e.  (TopOn `  B )  /\  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  /\  F : A --> B )  ->  (
y  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { x } )t  A ) ) `  F )  ->  y  e.  B ) )
5857ssrdv 3505 . . . . . . . . 9  |-  ( ( K  e.  (TopOn `  B )  /\  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  /\  F : A --> B )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  C_  B
)
5946, 54, 55, 58syl3anc 1223 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  C_  B
)
6043, 59syldan 470 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( cls `  J
) `  A )
)  ->  ( ( K  fLimf  ( ( ( nei `  J ) `
 { x }
)t 
A ) ) `  F )  C_  B
)
6142, 60syl5eqss 3543 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( cls `  J
) `  A )
)  ->  ran  ( { x }  X.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
6261ralrimiva 2873 . . . . 5  |-  ( ph  ->  A. x  e.  ( ( cls `  J
) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
63 iunss 4361 . . . . 5  |-  ( U_ x  e.  ( ( cls `  J ) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B 
<-> 
A. x  e.  ( ( cls `  J
) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
6462, 63sylibr 212 . . . 4  |-  ( ph  ->  U_ x  e.  ( ( cls `  J
) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
6538, 64syl5eqss 3543 . . 3  |-  ( ph  ->  ran  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
6637, 65eqsstrd 3533 . 2  |-  ( ph  ->  ran  ( ( JCnExt
K ) `  F
)  C_  B )
67 df-f 5585 . 2  |-  ( ( ( JCnExt K ) `
 F ) : C --> B  <->  ( (
( JCnExt K ) `
 F )  Fn  C  /\  ran  (
( JCnExt K ) `
 F )  C_  B ) )
6836, 66, 67sylanbrc 664 1  |-  ( ph  ->  ( ( JCnExt K
) `  F ) : C --> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374   E.wex 1591    e. wcel 1762   {cab 2447    =/= wne 2657   A.wral 2809    C_ wss 3471   (/)c0 3780   {csn 4022   <.cop 4028   U.cuni 4240   U_ciun 4320    X. cxp 4992   dom cdm 4994   ran crn 4995   Fun wfun 5575    Fn wfn 5576   -->wf 5577   ` cfv 5581  (class class class)co 6277   ↾t crest 14667   Topctop 19156  TopOnctopon 19157   clsccl 19280   neicnei 19359   Hauscha 19570   Filcfil 20076    fLimf cflf 20166  CnExtccnext 20289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-map 7414  df-pm 7415  df-rest 14669  df-fbas 18182  df-fg 18183  df-top 19161  df-topon 19164  df-cld 19281  df-ntr 19282  df-cls 19283  df-nei 19360  df-haus 19577  df-fil 20077  df-fm 20169  df-flim 20170  df-flf 20171  df-cnext 20290
This theorem is referenced by:  cnextcn  20297  cnextfres  20298
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